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An Exploratory Study of the Butterfly Effect Using Agent­Based Modeling

Mahmoud T. Khasawneh, Ph.D. Student, mkhasawn@odu.eduJun Zhang, Ph.D. Student, jxzhang@odu.edu

Nevan E. N. Shearer, Ph.D. Student, nesheare@odu.eduRaed M. Jaradat, Ph.D. Student, rjaradat@odu.edu

Elkin Rodriguez-Velasquez, Ph.D. Student, erodrig@odu.eduShannon R. Bowling, Assistant Professor, sbowling@odu.edu

Department of Engineering Management and Systems EngineeringOld Dominion University, Norfolk, Virginia 23529, USA

Abstract. This paper provides insights about the behavior of chaotic complex systems, and the sensitivedependence of the system on the initial starting conditions. How much does a small change in the initialconditions of a complex system affect it in the long term? Do complex systems exhibit what is called the"Butterfly Effect"? This paper uses an agent-based modeling approach to address these questions. Anexisting model from NetLogo© library was extended in order to compare chaotic complex systems withnear-identical initial conditions. Results show that small changes in initial starting conditions can have ahuge impact on the behavior of chaotic complex systems.

1. INTRODUCTION

The term the "butterfly effect" is attributed to the workof Edward Lorenz [1]. It is used to describe thesensitive dependence of the behavior of chaoticcomplex systems on the initial conditions of thesesystems. The metaphor refers to the notion that abutterfly flapping its wings somewhere may causeextreme changes in the ecological system's behavior inthe future, such as a hurricane.

2. LITERATURE REVIEW

Lorenz is major contributor to the concept of thebutterfly effect. He concluded that slight differing initialstates can evolve into considerably different states.Bewley [2] talked about the high sensitivity observed innonlinear complex systems, such as fluid convection,to very small levels of external force. Wang et al. [3]explored an approach for identifying chaoticphenomena in demands, and studied how a small driftin predicting an initial demand ultimately may cause asignificant difference to real demand. Palmer [4]argued that a hypothetical dynamically-unconstrainedperturbation to a small-scale variable, leaving all otherlarge-scale variables unchanged, would take thesystem in a completely different direction, off theattractor. Yugay and Yashkevich [5] mentioned that thebutterfly effect occurs in Long Josephson Junctions(LJJs) as described by a time dependent nonlinearsine-Gordon equation. This equation states that anyalteration within the initial perturbation fundamentallychanges the asymptotic state of the system. Socialsystems can also exhibit the butterfly effect

phenomenon. Several studies were dedicated toexamine the butterfly effect which resulted from theformat of the ballots in Palm Beach County, Floridaduring the presidential elections in the year 2000 [6, 7,8]. The chaos emerging from the confusingconfiguration of the dual-column ballot is said to havecaused 2,000 Democratic voters, a number larger thanthen Texas Governor George W. Bush's certifiedmargin of victory in Florida, to cast their vote foranother candidate instead of then Vice President AIGore, which effectively made George W. Bush the 43rdPresident of the United States.

3. METHODOLOGY AND MODELDEVELOPMENT

A modified version of the GasLab© model from thechemistry and physics library in etLogo© was used as abasis for our analysis of chaotic complex systems. Thefollowing are the assumptions of the modified GasLab©model:

• A random seed sets the initial conditions (x-ycoordinates, speed heading).

• Two types of agents: particles and diablos (the twoagents are identical, with the exception of name andcolor).

• The two types of agents only interact with theirown type. They do not interact with each other.

• For the complete duration of the simulation,particles are in blue, while diablos are in red.

• Agents move in a random heading and certainspeed until they collide with another agent of

https://ntrs.nasa.gov/search.jsp?R=20100012868 2020-05-27T23:08:19+00:00Z

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the same kind. Upon collision, a new speedand heading for the participatingparticles/diablos are set.

• Particles and diablos bounce off a wall andcontinue moving in the box.

• A collision occurs if two particles or two diablosare on the same patch.

The criterion this paper adopted to test the existence ofa butterfly effect is the average distance betweenparticles and diablos at each tick. The formula foraverage distance (D) is shown below:

N

I~(XAI -XBJ 2 + (YA1 -YBi /

D = ....:./=..:='------------

N(1 )

• N: number of particles/diablos in the system(for N = 10, 20, 30, ... , 100, 200, 300, ... ,1000) thus obtaining 19 configurations in total

• XAi : the x-coordinate of particle i• XBi : the x-coordinate of diablo i• YAi: the y-coordinate of particle i• YBi: the y-coordinate of diablo i

The reason behind using different numbers ofparticles/diablos is to examine the effect of the size ofthe population on the speed at which the butterfly effectemerges in the model.

The modeling methodology was divided into fivephases:

1. Creating a model with two random systems:The original GasLab© model had only oneagent; particles. Another agent, diablos, wasadded to the model with identical behaviorpatterns to those of particles. Because tworandom systems are created, particles use adifferent random seed than diablos (for speed,positioning, and heading).

2. Creating a model with same settings: After theestablishment of a model with two randomsystems, we then modified the model againso that particles and diablos use the samerandom seed, thus sharing the same speed,initial positioning, and heading, creating amodel with same initial settings. This modelwas the basis to test the hypothesis of thebutterfly effect. The rationale this paper usedto have a slider bar to incorporate extremelysmall changes to the heading of a singleagent, which we randomly chose to be a

diablo. Our assumption is that this smallchange is an equivalent to a butterfly "flappingits wings."

3. Automation setup for data collection: the codewas adjusted to avoid the need for doingmanual runs and to enable collection ofsufficient data to test the existence, or lackthereof, of a butterfly effect in the system. Alldata points were exported to a text file.

4. Statistical analysis: a macro was developed toorganize the data into an excel spreadsheet inorder to make the graphs and plot confidenceintervals.

5. Visual demonstration of divergence: aseparate model was created to visuallydemonstrate the point at which particle i anddiablo i diverge after starting in the sameposition. For the purpose of visualdemonstration, when the distance betweenparticle i and diablo i is equal to half-patch,their colors are changed to black to symbolizethe transition from identical systems torandom systems.

In recognition of the importance of systems' complexityin determining the existence of a butterfly effect, we ranour model(s) for different configurations ofparticles/diablos as mentioned earlier. Moreover, toreduce the effect of randomness and obtain confidenceintervals for our results, each configuration was run for30 times, each run consisting of 10,000 ticks.

4. DISCUSSION AND CONCLUSIONS

Figures 1, 2, and 3 show the average distancebetween two random systems for 10, 500, and 1000particles. The graphs illustrate that regardless of thenumber of agents we have in the model, the averagedistance tends to fluctuate around 38. It is evident thatthe variance decreases as the number of agentsincreases.

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Figure 5: Average Distance for 500 Particles/Diablos(Same Settings)

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Figure 4: Average Distance for 10 Particles/Diablos(Same Settings)

Figures 4, 5, and 6 show the average distancebetween two systems with same settings for 10, sao,and 1000 particles/diablos, with the exception ofmaking a change to the heading of one diablo toexamine the butterfly effect. An observation is that asthe number of particles/diablos is increased in themodel, the system diverges quicker. Similarly to therandom systems, as the number of particles/diablos isincreased, the variance decreases.

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Figure 6: Average Distance for 1000 ParticieslDiablos(Same Settings)

Figures 7, 8, and 9 show the difference in averagedistance between the model with random systems andthe model with same settings. In all cases, the modelwith same settings will approach the same conditionsas the model with random systems. Although themodel with same settings quickly approaches thebehavior of the model with random systems, it takeslonger to actually reach the same average distance of38. Moreover, as the number of particles/diablosincreases, it takes longer to reach the same averagedistance for model of random systems.

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The most important implication of this study is thatchaotic complex systems can actually exhibit thebutterfly effect. Scientists, from all disciplines, shouldacknowledge that when studying complex systems andcomplex phenomena, reaching an understanding of thecurrent state of the systems can be traced back to asmall perturbation earlier in the system's life cycle.

1000 hrticle. one! ~.bIos (Difference)

Figure 8: Average Distance for 500 Particles/Diablos(Difference)

Figure 9: Average Distance for 1000 Particles/Diablos(Difference)

REFERENCES1. Lorenz, E., 1963, "Deterministic Nonperiodic

Flow", Journal of The Atmospheric Sciences,20,130-141.

2. Bewley, T. R., 1999, "Linear control andestimation of nonlinear chaotic convection:Harnessing the butterfly effect", Physics ofFluid, 11(5), 1169-1186.

3. Wang, K. J., Wee, H.,M., Gao, S., F., andChung, S., L., 2004, "Production and inventorycontrol with chaotic demands", Omega Journalof Management Science, 33, 97- 106.

4. Palmer, T. N., 2005, "Quantum Reality,Complex Numbers, and The MeteorologicalButterfly Effect", Bulletin of the AmericanMeteorological Society, 86(4), 519-530.

5. Yugay, K. N. and Yashkevich, E. A 2006, "TheBradbury Butterfly Effect in Long JosephsonJunctions", Journal of Superconductivity andNovel Magnetism, 19(1/2), 1-9.

6. Sinclair, R. C., Mark, M. M., Moore, S. E.,Lavis, C. A, and Soldat, AS., 2000,"Psychology: An electoral butterfly effect",Nature, 408, 665-666.

7. Wand, J., Shotts, K., Sekhon, J., Mebane Jr.,W., Herron, M., Brady, H., 2001, "The ButterflyDid it: Aberrant Vote for Buchanan in PalmBeach County, Florida", American PoliticalScience Review. 95 (4),793-810.

8. Wu, D.W., 2002, "Regression Analyses On theButterfly Ballot Effect: A Statistical Perspectiveof the US 2000 Election", International Journalof Mathematical Education in Science andTechnology, 33, 309-317.

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Collectively, the results of this paper demonstrate thatthere is a butterfly effect in chaotic complex systems. Infact, as complexity increases, the butterfly effectemerges quicker but takes a longer time to completelyreplicate the model with random systems. Therefore,an additional experiment was run to determine howlong it takes for the model with same settings tocompletely replicate the model with random systems.As evident in Figure 10, the results of the modelindicate that it actually takes about 2 million ticks tocompletely replicate the model of random systems, forthe setting of 1000 particles/diablos.

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Figure 10: Simulation Results for 1000Particles/Diablos After 2 Million Ticks